Finite Group Databasebeta

← #15799633

Subgroups — #15799633

Order 3328.

1934 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#88123C211yesmaximal, characteristic
2C2×#11718640C214yesmaximal
3C2×#11718637C212yesmaximal
4#13113100C214yesmaximal
5#13113065C212yesmaximal
6C4×#88071C211yesmaximal, characteristic
7C2×C13×#389C211yesmaximal, characteristic
8C23×Dic26C2212yes—
9C22×C13×C4⋊C4C2211yescharacteristic
10C22×C13×C4⋊C4C2212yes—
11C22×C13×C4⋊C4C2211yescharacteristic
12C22×C42×C13C2211yescharacteristic
13C22×C4⋊Dic13C2212yes—
14C22×C4⋊Dic13C2212yes—
15C22×#6101C2218yes—
16C22×C4×Dic13C2212yes—
17C2×#88123—24no—
18C2×#88123—28no—
19C13×#389—24no—
20C2×C13×#95C2212yes—
21C2×#88071C2218yes—
22C2×#88071C2212yes—
23C2×#88071C2212yes—
24#11718640—216no—
25#11718637—28no—
26C23×C4×C13C2312yes—
27C23×C4×C13C2312yes—
28C23×C4×C13C2311yescharacteristic
29C23×C4×C13C2311yescharacteristic
30C23×C4×C13C2311yescharacteristic
31C23×Dic13C2314yes—
32C23×Dic13C2314yes—
33C22×Dic26—24no—
34C22×Dic26—28no—
35C22×Dic26—28no—
36C22×Dic26—44no—
37C2×C13×C4⋊C4—22no—
38C2×C13×C4⋊C4—22no—
39C2×C13×C4⋊C4—22no—
40C2×C13×C4⋊C4—24no—
41C2×C13×C4⋊C4—24no—
42C2×C13×C4⋊C4—28no—
43C2×C13×C4⋊C4Q814yescomplemented
44C2×C42×C13—24no—
45C2×C42×C13D414yes—
46C2×C4⋊Dic13—24no—
47C2×C4⋊Dic13—24no—
48C2×C4⋊Dic13—24no—
49C2×C4⋊Dic13—28no—
50C2×C4⋊Dic13D418yes—
51C2×#6101—216no—
52C2×#6101—216no—
53C2×#6101—48no—
54C2×C4×Dic13—24no—
55C2×C4×Dic13—24no—
56C2×C4×Dic13Q818yescomplemented
57#88123—232no—
58C13×#95—24no—
59#88071—216no—
60#88071D418yes—
61#88071Q818yescomplemented
62#15143—131nomaximal, Sylow
63C24×C13C2411yescharacteristic
64C22×C4×C13—22no—
65C22×C4×C13—22no—
66C22×C4×C13—22no—
67C22×C4×C13—22no—
68C22×C4×C13—24no—
69C22×C4×C13—22no—
70C22×C4×C13—22no—
71C22×C4×C13—24no—
72C22×C4×C13—22no—
73C22×C4×C13—24no—
74C22×C4×C13—24no—
75C22×C4×C13—24no—
76C22×C4×C13C2×D414yes—
77C22×C4×C13C2×D414yes—
78C22×C4×C13C2×D412yes—
79C22×C4×C13C2×D412yes—
80C22×C4×C13C2×D412yes—
81C22×C4×C13C2×Q814yes—
82C22×C4×C13C2×Q812yes—
83C22×C4×C13C2×Q814yes—
84C22×C4×C13C2×Q814yes—
85C22×C4×C13#4112yes—
86C22×Dic13—24no—
87C22×Dic13—24no—
88C22×Dic13—24no—
89C22×Dic13—24no—
90C22×Dic13—28no—
91C22×Dic13—24no—
92C22×Dic13—24no—
93C22×Dic13—28no—
94C22×Dic13C2×D418yes—
95C22×Dic13C2×D418yes—
96C22×Dic13C2×Q818yes—
97C22×Dic13C2×Q818yes—
98C2×Dic26—216no—
99C2×Dic26—432no—
100C2×Dic26—48no—
101C2×Dic26—48no—
102C13×C4⋊C4—28no—
103C13×C4⋊C4—44no—
104C13×C4⋊C4—48no—
105C42×C13—28no—
106C4⋊Dic13—216no—
107C4⋊Dic13—216no—
108#6101—432no—
109C4×Dic13—216no—
110C22×C4⋊Q8—131no—
111C2×#395—134no—
112C2×#389D1311yescharacteristic
113C2×#384—132no—
114#1444—134no—
115C4⋊#95—132no—
116C4×#95—131no—
117C23×C13C22×D411yescharacteristic
118C23×C13C22×D412yes—
119C23×C13C22×D412yes—
120C23×C13C22×Q812yes—
121C23×C13C22×Q811yescharacteristic
122C23×C13C22×Q812yes—
123C23×C13C2×#4111yescharacteristic
124C23×C13H(5,𝔽2)12yes—
125C23×C132−1+412yes—
126C2×C4×C13—22no—
127C2×C4×C13—22no—
128C2×C4×C13—22no—
129C2×C4×C13—22no—
130C2×C4×C13—28no—
131C2×C4×C13—28no—
132C2×C4×C13—24no—
133C2×C4×C13—28no—
134C2×C4×C13—42no—
135C2×C4×C13—42no—
136C2×C4×C13—44no—
137C2×C4×C13—42no—
138C2×C4×C13—42no—
139C2×C4×C13—42no—
140C2×C4×C13—48no—
141C2×C4×C13#12014yes—
142C2×C4×C13C22⋊Q818yes—
143C2×C4×C13C22⋊Q814yes—
144C2×C4×C13#12714yes—
145C2×C4×C13C4⋊Q814yes—
146C2×C4×C13C4⋊Q818yes—
147C2×C4×C13C4⋊Q814yes—
148C2×Dic13—28no—
149C2×Dic13—28no—
150C2×Dic13—28no—
151C2×Dic13—28no—
152C2×Dic13—28no—
153C2×Dic13—28no—
154C2×Dic13—28no—
155C2×Dic13—216no—
156C2×Dic13—44no—
157C2×Dic13—44no—
158C2×Dic13—48no—
159C2×Dic13C4⋊Q8116yescomplemented
160Dic26—432no—
161C23×Q8—132no—
162C2×C4⋊Q8—268no—
163C2×C4⋊Q8—264no—
164C22×C4⋊C4—132no—
165C22×C4⋊C4—132no—
166C22×C4⋊C4—138no—
167C22×C4⋊C4C2×D1312yes—
168C22×C4⋊C4C2×D1311yescharacteristic
169C22×C4⋊C4C2×D1311yescharacteristic
170C22×C42—132no—
171C22×C42C2×D1311yescharacteristic
172#395—2616no—
173#389—24no—
174#384—268no—
175C2×#95—132no—
176C2×#95—132no—
177C2×#95—138no—
178C2×#95C2×D1312yes—
179C22×C13C2×#12011yescharacteristic
180C22×C13C2×C22⋊Q812yes—
181C22×C13C2×C22⋊Q811yescharacteristic
182C22×C13C2×#12711yescharacteristic
183C22×C13C2×C4⋊Q814yes—
184C22×C13C2×C4⋊Q812yes—
185C22×C13C2×C4⋊Q811yescharacteristic
186C22×C13C2×C4⋊Q811yescharacteristic
187C22×C13#53511yescharacteristic
188C22×C13#53611yescharacteristic
189C22×C13#54414yes—
190C22×C13D4214yes—
191C22×C13#54712yes—
192C22×C13D4×Q812yes—
193C22×C13D4×Q814yes—
194C22×C13#55412yes—
195C22×C13Q8212yes—
196C4×C13—24no—
197C4×C13—24no—
198C4×C13—44no—
199C4×C13—82no—
200C4×C13#39518yescomplemented
201Dic13—216no—
202Dic13—48no—
203C22×Q8—268no—
204C22×Q8—268no—
205C22×Q8—264no—
206C22×Q8—524no—
207C23×C4—134no—
208C23×C4—134no—
209C23×C4C22×D1312yes—
210C23×C4C22×D1312yes—
211C23×C4C22×D1311yescharacteristic
212C23×C4C22×D1311yescharacteristic
213C23×C4C22×D1311yescharacteristic
214C4⋊Q8—2632no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—138no—
222C2×C4⋊C4—268no—
223C2×C4⋊C4—264no—
224C2×C4⋊C4—264no—
225C2×C4⋊C4—264no—
226C2×C4⋊C4—2616no—
227C2×C4⋊C4—2616no—
228C2×C4⋊C4—528no—
229C2×C4⋊C4Dic2614yescomplemented
230C2×C42—24no—
231C2×C42—138no—
232C2×C42—264no—
233C2×C42—264no—
234C2×C42C22⋊D1314yes—
235#95—24no—
236#95—138no—
237#95—138no—
238#95—2616no—
239C2×C13C2×#39511yescharacteristic
240C2×C13#243311yescharacteristic
241C2×C13#243411yescharacteristic
242C2×C13#244014yes—
243C2×C13#254211yescharacteristic
244C2×C13#255114yes—
245C2×C13#256711yescharacteristic
246C2×C13#266011yescharacteristic
247C2×C13#286811yescharacteristic
248C24C23×D1311yescharacteristic
249C2×Q8—2616no—
250C2×Q8—5232no—
251C2×Q8—528no—
252C2×Q8—528no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—138no—
266C22×C4—138no—
267C22×C4—138no—
268C22×C4—138no—
269C22×C4—264no—
270C22×C4—264no—
271C22×C4—264no—
272C22×C4—264no—
273C22×C4—268no—
274C22×C4—264no—
275C22×C4—268no—
276C22×C4—264no—
277C22×C4C2×Dic2614yes—
278C22×C4C2×Dic2612yes—
279C22×C4#612712yes—
280C22×C4D4×D1314yes—
281C22×C4D4×D1314yes—
282C22×C4Q8×D1314yes—
283C22×C4Q8×D1314yes—
284C22×C4C2×C22⋊D1312yes—
285C22×C4C2×C22⋊D1312yes—
286C22×C4C2×C22⋊D1312yes—
287C4⋊C4—28no—
288C4⋊C4—44no—
289C4⋊C4—48no—
290C4⋊C4—2616no—
291C4⋊C4—2616no—
292C4⋊C4—5232no—
293C42—28no—
294C42—2616no—
295C13#1514311yescharacteristic, complemented, Sylow
296C23C22×Dic2611yescharacteristic
297C23C2×#612711yescharacteristic
298C23C2×D4×D1312yes—
299C23C2×D4×D1312yes—
300C23C2×Q8×D1312yes—
301C23C2×Q8×D1312yes—
302C23#8825612yes—
303C23#8825712yes—
304C23C22×C22⋊D1311yescharacteristic
305Q8—5232no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—28no—
311C2×C4—28no—
312C2×C4—24no—
313C2×C4—28no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—48no—
321C2×C4—1316no—
322C2×C4—268no—
323C2×C4—268no—
324C2×C4—268no—
325C2×C4—268no—
326C2×C4—268no—
327C2×C4—268no—
328C2×C4—268no—
329C2×C4—2616no—
330C2×C4—524no—
331C2×C4—524no—
332C2×C4—528no—
333C2×C4#8814218yes—
334C2×C4#8815018yes—
335C2×C4#8817814yes—
336C2×C4#8819114yes—
337C2×C4#8819414yes—
338C2×C4#8819814yes—
339C2×C4#8819814yes—
340C22C2×#8814212yes—
341C22C2×#8815012yes—
342C22D4×Dic2612yes—
343C22#1171925612yes—
344C22Q8×Dic2612yes—
345C22#1171928412yes—
346C22#1171931414yes—
347C22#1171933714yes—
348C22#1171939414yes—
349C22D13×C4⋊Q814yes—
350C22C2×#8817811yescharacteristic
351C22C2×#8819111yescharacteristic
352C22C2×#8819411yescharacteristic
353C22C2×#8819811yescharacteristic
354C22C2×#8819811yescharacteristic
355C22#1171953011yescharacteristic
356C22#1171953511yescharacteristic
357C4—24no—
358C4—24no—
359C4—44no—
360C4—82no—
361C4—2616no—
362C4—528no—
363C4#1171864018yescomplemented
364C2C2×#1171864011yescharacteristic
365C2#1311847411yescharacteristic
366C2#1311848111yescharacteristic
367C2#1311854014yes—
368C2#1311978911yescharacteristic
369C2#1311986414yes—
370C2#1312004111yescharacteristic
371C2#1312095711yescharacteristic
372C2#1312323211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D13
commutator, socle, prime radical, cosocleC24×C13C24
FittingC2×C13×#389C2
hypercenterC2×#389D13
hypocenterC13#15143
LCSG ▷ C24×C13 ▷ C13
UCSC1 ◁ C24 ◁ C2×#389

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
43. C2×C13×C4⋊C4Q8semidirect
56. C2×C4×Dic13Q8semidirect
61. #88071Q8semidirect
159. C2×Dic13C4⋊Q8semidirect
200. C4×C13#395semidirect
229. C2×C4⋊C4Dic26semidirect
295. C13#15143semidirect
363. C4#11718640semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#881232
C2×#117186402
C2×#117186372
#131131002
#131130652
C4×#880712
C2×C13×#3892
#1514313